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\begin{document}

\begin{center}
{\LARGE Trade Flows, Multilateral Resistance, and Firm Heterogeneity}

By Alberto Behar and Benjamin Nelson
\end{center}

\appendix


\section{Technical appendix}

\subsection{Theory}

\subsubsection{Derivation of gravity equation}

HMR's set-up yield imports by country $i$ from country $j$ of:%
\begin{equation}
M_{ij}=\left(  \frac{c_{j}t_{ij}}{\alpha P_{i}}\right)  ^{1-\sigma}N_{j}%
Y_{i}V_{ij}, \label{HMR06}%
\end{equation}
(cf. HMR equation (6)). Assuming trade balance yields $Y_{j}=\sum_{i\in I_{j}%
}M_{ij}$, such that summing both sides of HMR's expression gives:%
\begin{equation}
Y_{j}=\left(  \frac{c_{j}}{\alpha}\right)  ^{1-\sigma}N_{j}\sum_{i\in I_{j}%
}\left(  \frac{t_{ij}}{P_{i}}\right)  ^{1-\sigma}Y_{i}V_{ij}\Rightarrow\left(
\frac{c_{j}}{\alpha}\right)  ^{1-\sigma}N_{j}=\frac{Y_{j}}{\sum_{i\in I_{j}%
}\left(  \frac{t_{ij}}{P_{i}}\right)  ^{1-\sigma}Y_{i}V_{ij}} \label{costs}%
\end{equation}
Using this expression to substitute factor costs from equation (\ref{HMR06})
then yields bilateral imports of:%
\[
M_{ij}=Y_{i}Y_{j}\frac{1}{\sum_{i\in I_{j}}\left(  \frac{t_{ij}}{P_{i}%
}\right)  ^{1-\sigma}Y_{i}V_{ij}}\left(  \frac{t_{ij}}{P_{i}}\right)
^{1-\sigma}V_{ij}.
\]
Dividing and multiplying by $Y^{I_{j}}$, the combined incomes of all importers
from country $j$, $Y^{I_{j}}\equiv\sum_{i\in I_{j}}Y_{i}$, then gives:%
\begin{equation}
M_{ij}=\frac{Y_{i}Y_{j}}{Y^{I_{j}}}\left(  \frac{t_{ij}}{P_{i}\widehat{P}_{j}%
}\right)  ^{1-\sigma}V_{ij}, \label{gravity}%
\end{equation}
\qquad where%
\begin{equation}
\widehat{P}_{j}^{1-\sigma}\equiv\sum_{i\in I_{j}}\left(  \frac{t_{ij}}{P_{i}%
}\right)  ^{1-\sigma}s_{i}^{I_{j}}V_{ij} \label{phat}%
\end{equation}
is outward multilateral resistance, in which we define $s_{i}^{I_{j}}\equiv
Y_{i}/Y^{I_{j}}$. The expression for $P_{i}$, or inward multilateral
resistance, follows from the definition of the CES price index for consumers
in country $i$. This is given by:%
\[
P_{i}^{1-\sigma}=\int_{\gamma\in\Gamma_{i}}p_{i}(\gamma)^{1-\sigma}d\gamma
\]
(cf. HMR\ equation 2) where $p_{i}(\gamma)$ is the price faced by consumers in
$i$ of product variety $\gamma$ in the set of products consumed in $i$,
$\Gamma_{i}$. By monopolistic competition, firms set prices such that
$p_{i}(a)=t_{ij}c_{j}a/\alpha$, where $a$ is the (inverse) productivity draw
of firm of type $a$, and $\alpha\equiv1-(1/\sigma)$ is a transformation of the
elasticity of substitution, capturing the mark-up. Using this in the price
index for all import origins yields:%
\[
P_{i}^{1-\sigma}=\sum_{j\in J_{i}}\left(  \frac{t_{ij}c_{j}}{\alpha}\right)
^{1-\sigma}N_{j}V_{ij}%
\]
since each exporter $j$ has measure $N_{j}$ firms and where $V_{ij}\equiv
\int_{a_{L}}^{a_{ij}}a^{1-\sigma}dG(a)$, in which $a_{L}$ is the lower bound
on the support of (inverse) productivity draws, which has c.d.f. $G(a)$. Using
equation (\ref{costs}) then gives $P_{i}^{1-\sigma}=\sum_{j\in J_{i}}\left(
t_{ij}/\widehat{P}_{j}\right)  ^{1-\sigma}s_{j}^{J_{i}}r_{ij}V_{ij}$, where
$s_{j}^{J_{i}}\equiv Y_{j}/Y^{J_{i}}$ and where $r_{ij}\equiv Y^{J_{i}%
}/Y^{I_{j}}$. The latter is a scaling factor that increases as the collective
size of exporters to $i$ grows larger relative to the size of importers from
$j$. Recall that a higher $P_{i}$ encourages more trade between $i$ and $j$,
as it reflects higher inward multilateral resistance. When $r_{ij}$ is high on
average, $P_{i}$ tends to be low, discouraging exports from $j$ to $i$. Since
$r_{ij}\equiv\left(  \sum_{j\in J_{i}}Y_{j}\right)  /\left(  \sum_{i\in I_{j}%
}Y_{i}\right)  $ is a constant for pair $i-j$ when the sets of active traders
$\{I_{j},J_{i}\}$ are constant, and we hold these sets constant in our
comparative statics (justified by the empirical result that little of the
expansion of trade is due to country entry -- see HMR and Table 5 of the
text), we can discard it for comparative statics purposes, and use:%
\begin{equation}
P_{i}^{1-\sigma}=\sum_{j\in J_{i}}\left(  \frac{t_{ij}}{\widehat{P}_{j}%
}\right)  ^{1-\sigma}s_{j}^{J_{i}}V_{ij}, \label{p}%
\end{equation}
\qquad Equations (\ref{gravity}), (\ref{p}) and (\ref{phat}) correspond to
equations (1), (2) and (3) in the text.

The gravity equation for the extensive margin (equation 4 in the text) is
derived as follows. As in HMR, a firm in country $j$ producing variety
$\gamma$ faces demand of $x_{j}(\gamma)=\left(  Y_{j}/P_{j}^{1-\sigma}\right)
p_{j}(\gamma)^{-\sigma}$. A firm with (inverse)\ productivity $a$ in country
$j$ serving consumers in $i$ has a cost function $C_{ij}(a)=at_{ij}c_{j}%
x_{i}(\gamma)+c_{j}f_{ij}$. Using the firm's profit-maximising resulting price
$p_{i}(a)=t_{ij}c_{j}a/\alpha$, the profit for a firm with productivity $a$
exporting from $j$ to $i$ is:%
\[
\pi_{ij}(a)=(1-\alpha)\left(  \frac{t_{ij}c_{j}a}{\alpha P_{i}}\right)
^{1-\sigma}Y_{i}-c_{j}f_{ij}.
\]
Now define a cut-off level of (inverse)\ productivity $a_{ij}$ such that a
firm with this productivity makes zero profit. The cut-off is given by:%
\[
a_{ij}=\left[  \frac{\left(  1-\alpha\right)  Y_{i}}{c_{j}f_{ij}}\right]
^{\frac{1}{\sigma-1}}\frac{\alpha P_{i}}{t_{ij}c_{j}},
\]
(see HMR equation 4). Next, use (\ref{costs}) above to write%
\[
c_{j}=\alpha\left(  \frac{Y^{I_{j}}N_{j}\widehat{P}_{j}^{1-\sigma}}{Y_{j}%
}\right)  ^{\frac{1}{\sigma-1}},
\]
such that the cut-off can be written as:%
\[
a_{ij}^{\sigma-1}=\frac{\left(  1-\alpha\right)  }{N_{j}}\frac{Y_{i}Y_{j}%
}{Y^{I_{j}}}\frac{1}{f_{ij}}\left(  \frac{t_{ij}}{P_{i}\widehat{P}_{j}%
}\right)  ^{1-\sigma},
\]
which is the gravity equation for the extensive margin given by equation 4 in
the text.

\subsubsection{Proof of Lemma 1: Effect of country size on extensive margin
elasticity}

\begin{proof}
For $h=i,j$,%
\begin{align*}
\left.  \frac{\partial\varphi_{ij}}{\partial y_{h}}\right\vert _{P_{i},\hat
{P}_{j}}  &  =\left.  \frac{\partial}{\partial y_{h}}\frac{\delta e^{\delta
z_{ij}}}{e^{\delta z_{ij}}-1}\right\vert _{P_{i},\hat{P}_{j}}\\
&  =\frac{\delta e^{\delta z_{ij}}}{e^{\delta z_{ij}}-1}\left(  \delta
-\frac{\delta e^{\delta z_{ij}}}{e^{\delta z_{ij}}-1}\right)  \left.
\frac{\partial z_{ij}}{\partial y_{h}}\right\vert _{P_{i},\hat{P}_{j}}\\
&  =-\frac{\delta^{2}e^{\delta z_{ij}}}{\left(  e^{\delta z_{ij}}-1\right)
^{2}}<0,
\end{align*}
since $\left.  \partial z_{ij}/\partial y_{h}\right\vert _{P_{i},\hat{P}_{j}%
}=1$ for $h=i,j$. For fixed world output and given active traders, the
elasticity of $\varphi_{ij}$\ with respect to (log) GDP will have the same
sign as the elasticity of $\varphi_{ij}$\ with respect to GDP\ shares, since
shares $s_{h}^{X_{h}}\equiv Y_{h}/Y^{X_{h}},$\ $h=i,j,$\ $X=J,I$\ are
trivially increasing in output levels $Y_{h}$. Therefore:%
\[
\frac{\partial\varphi_{ij}}{\partial s_{h}}<0\text{, \ \ }h=i,j.
\]

\end{proof}

\subsubsection{Taylor approximation\label{App_MRapprox}}

Following Baier and Bergstrand, we take a first-order Taylor approximation of
the MR indices. Our model contains an additional effect due to the impact of
MR\ on the extensive margin: the price indices contain extensive margin terms,
which themselves contain a system of price indices. In order to extract the
exogenous components of the system, we make the further assumption that the
extensive margin terms that appear in the price index equations (but not the
main gravity equation) can be decomposed according to $V_{ij}\simeq
\widetilde{Z}_{ij}^{\delta}\left(  P_{i}\widehat{P}_{j}\right)  ^{\delta
(\sigma-1)}$, or $w_{ij}\simeq\delta\widetilde{z}_{ij}+\ln\left(
P_{i}\widehat{P}_{j}\right)  ^{\delta(\sigma-1)}$. This approximated extensive
margin value approaches that computed under the assumption of a Pareto
distribution for $\delta z_{ij}$ sufficiently large, since $w_{ij}=\ln\left\{
e^{\delta z_{ij}}-1\right\}  \simeq\delta z_{ij}=\delta\widetilde{z}_{ij}%
+\ln\left(  P_{i}\widehat{P}_{j}\right)  ^{\delta(\sigma-1)}$ when this is the
case. Using this, and defining $W_{ij}\equiv\widetilde{Z}_{ij}^{\delta}$
yields price indices of%
\begin{align}
\widehat{P}_{j}^{-k}  &  =\sum\limits_{i\in I_{j}}t_{ij}^{1-\sigma}%
s_{i}^{I_{j}}P_{i}^{k}W_{ij}\label{pj}\\
P_{i}^{-k}  &  =\sum\limits_{j\in J_{i}}t_{ij}^{1-\sigma}s_{j}^{J_{i}%
}\widehat{P}_{j}^{k}W_{ij}, \label{pi}%
\end{align}
where we have used that $(\delta+1)(\sigma-1)=(\frac{k-\sigma+1}{\sigma
-1}+1)(\sigma-1)=k$. The terms (\ref{pj}) and (\ref{pi}) are those which we
Taylor approximate. We do this around a symmetric centre in which trade
frictions are positive but symmetric and sufficiently low for all countries to
trade. Then $t_{ij}=t>1$ for all $i,j$. Similarly, $W_{ij}=W$, $P_{i}=P$,
$I_{j}=J_{i}$ and $\widehat{P}_{j}=\widehat{P}$. This yields the system%
\begin{align*}
\widehat{P}^{-k}  &  =t^{1-\sigma}P^{k}W\sum\limits_{i}s_{i}=t^{1-\sigma}%
P^{k}W,\\
P^{-k}  &  =t^{1-\sigma}\widehat{P}^{k}W\sum\limits_{j}s_{j}=t^{1-\sigma
}\widehat{P}^{k}W,
\end{align*}
which has a solution at $P=\widehat{P}$. Any observed variations in $t_{ij}$
are pertubations from this centre which can be approximated using Taylor's
method. These can be asymmetric in the sense that $t_{ij}\neq t_{ji}$. To do
this, we use%
\begin{align}
e^{-k\ln\widehat{P}_{j}}  &  =\sum\limits_{i\in I_{j}}e^{\ln s_{i}^{I_{j}}%
}e^{(1-\sigma)\ln t_{ij}}e^{k\ln P_{i}}e^{\ln W_{ij}},\\
e^{-k\ln P_{i}}  &  =\sum\limits_{j\in J_{i}}e^{\ln s_{j}^{J_{i}}}%
e^{(1-\sigma)\ln t_{ij}}e^{k\ln\widehat{P}_{j}}e^{\ln W_{ij}}, \label{MR_pi}%
\end{align}
Then expanding the LHS of (\ref{MR_pi}) around $\ln P$ and the RHS around $\ln
t$, $\ln W$, and $\ln\widehat{P}$ gives%
\begin{multline*}
P^{-k}-kP^{-k}[\ln P_{i}-\ln P]=\sum_{j\in J_{i}}s_{j}^{J_{i}}t^{1-\sigma
}W\widehat{P}^{k}+\sum_{j\in J_{i}}s_{j}^{J_{i}}(1-\sigma)t^{1-\sigma
}W\widehat{P}^{k}[\ln t_{ij}-\ln t]\\
+\sum_{j\in J_{i}}s_{j}^{J_{i}}t^{1-\sigma}W\widehat{P}^{k}[\ln W_{ij}-\ln
W]+\sum_{j\in J_{i}}s_{j}^{J_{i}}kt^{1-\sigma}W\widehat{P}^{k}[\ln
\widehat{P}_{j}-\ln\widehat{P}].
\end{multline*}
Divide through by $P^{-k}=t^{1-\sigma}W\widehat{P}^{k}$ and use that
$\sum_{j\in J_{i}}s_{j}^{J_{i}}=1$ to give%
\begin{multline*}
-k[\ln P_{i}-\ln P]=\sum_{j\in J_{i}}s_{j}^{J_{i}}(1-\sigma)[\ln t_{ij}-\ln
t]\\
+\sum_{j\in J_{i}}s_{j}^{J_{i}}[\ln W_{ij}-\ln W]+\sum_{j\in J_{i}}%
s_{j}^{J_{i}}k[\ln\widehat{P}_{j}-\ln\widehat{P}]\\
=\sum_{j\in J_{i}}s_{j}^{J_{i}}(1-\sigma)\ln t_{ij}-(1-\sigma)\ln t\\
+\sum_{j\in J_{i}}s_{j}^{J_{i}}\ln W_{ij}-\ln W+\sum_{j\in J_{i}}s_{j}^{J_{i}%
}k\ln\widehat{P}_{j}-k\ln\widehat{P}.
\end{multline*}
Use $-k\ln P=(1-\sigma)\ln t+\ln W+k\ln\widehat{P}$ to give%
\[
\ln P_{i}=\frac{1}{k}\sum_{j\in J_{i}}s_{j}^{J_{i}}\left[  \left(
\sigma-1\right)  \ln t_{ij}-\ln W_{ij}\right]  -\sum_{j\in J_{i}}s_{j}^{J_{i}%
}\ln\widehat{P}_{j}.
\]
We can obtain the following expression for exporter resistance analogously:%
\[
\ln\widehat{P}_{j}=\frac{1}{k}\sum_{i\in I_{j}}s_{i}^{I_{j}}\left[  \left(
\sigma-1\right)  \ln t_{ij}-\ln W_{ij}\right]  -\sum_{i\in I_{j}}s_{i}^{I_{j}%
}\ln P_{i}.
\]
Substituting the first of these into the second yields:%
\begin{multline*}
\ln\widehat{P}_{j}=\frac{1}{k}\sum_{i\in I_{j}}s_{i}^{I_{j}}\left[  \left(
\sigma-1\right)  \ln t_{ij}-\ln W_{ij}\right] \\
-\sum_{i\in I_{j}}s_{i}^{I_{j}}\left[  \frac{1}{k}\sum_{j\in J_{i}}%
s_{j}^{J_{i}}\left[  \left(  \sigma-1\right)  \ln t_{ij}-\ln W_{ij}\right]
-\sum_{j\in J_{i}}s_{j}^{J_{i}}\ln\widehat{P}_{j}\right]  ,
\end{multline*}
such that adding $\ln P_{i}+$ $\ln\widehat{P}_{j}$ gives:%
\begin{multline*}
\ln P_{i}\widehat{P}_{j}=-\frac{1}{k}\sum_{i\in I_{j}}s_{i}^{I_{j}}\sum_{j\in
J_{i}}s_{j}^{J_{i}}\left[  \left(  \sigma-1\right)  \ln t_{ij}-\ln
W_{ij}\right] \\
+\frac{1}{k}\sum_{j\in J_{i}}s_{j}^{J_{i}}\left[  \left(  \sigma-1\right)  \ln
t_{ij}-\ln W_{ij}\right]  +\frac{1}{k}\sum_{i\in I_{j}}s_{i}^{I_{j}}\left[
\left(  \sigma-1\right)  \ln t_{ij}-\ln W_{ij}\right]  .
\end{multline*}
Writing the approximated price indices in terms using subscript $h$ as the
generic exporter and $l$ as the generic importer, we have%
\begin{multline*}
\ln P_{i}\widehat{P}_{j}=-\frac{1}{k}\sum_{l\in I_{j}}s_{l}^{I_{j}}\sum_{h\in
J_{i}}s_{h}^{J_{i}}\left[  \left(  \sigma-1\right)  \ln t_{lh}-\ln
W_{lh}\right] \\
+\frac{1}{k}\sum_{h\in J_{i}}s_{h}^{J_{i}}\left[  \left(  \sigma-1\right)  \ln
t_{ih}-\ln W_{ih}\right]  +\frac{1}{k}\sum_{l\in I_{j}}s_{l}^{I_{j}}\left[
\left(  \sigma-1\right)  \ln t_{lj}-\ln W_{lj}\right]  .
\end{multline*}
Using $\left(  \sigma-1\right)  \ln t_{ij}=\gamma d_{ij}$ and $\ln
W_{ij}=\delta\widetilde{z}_{ij}$, our MR\ approximation is then given by%
\[
\ln P_{i}\widehat{P}_{j}=-\frac{1}{k}\sum_{l\in I_{j}}s_{l}^{I_{j}}\sum_{h\in
J_{i}}s_{h}^{J_{i}}\left[  \gamma d_{lh}-\delta\widetilde{z}_{lh}\right]
+\frac{1}{k}\sum_{h\in J_{i}}s_{h}^{J_{i}}\left[  \gamma d_{ih}-\delta
\widetilde{z}_{ih}\right]  +\frac{1}{k}\sum_{l\in I_{j}}s_{l}^{I_{j}}\left[
\gamma d_{lj}-\delta\widetilde{z}_{lj}\right]  .
\]
Our comparative statics require the term $\frac{\partial\ln\left(  P_{i}%
P_{j}\right)  ^{\sigma-1}}{\partial d}=(\sigma-1)\frac{\partial\ln\left(
P_{i}P_{j}\right)  }{\partial d}$, so we write%
\begin{multline*}
(\sigma-1)\ln P_{i}\widehat{P}_{j}\\
=\frac{(\sigma-1)}{k}\left\{  -\sum_{l\in I_{j}}s_{l}^{I_{j}}\sum_{h\in J_{i}%
}s_{h}^{J_{i}}\left[  \gamma d_{lh}-\delta\widetilde{z}_{lh}\right]
+\sum_{h\in J_{i}}s_{h}^{J_{i}}\left[  \gamma d_{ih}-\delta\widetilde{z}%
_{ih}\right]  +\sum_{l\in I_{j}}s_{l}^{I_{j}}\left[  \gamma d_{lj}%
-\delta\widetilde{z}_{lj}\right]  \right\}  ,
\end{multline*}
and use $\delta=\frac{k-\sigma+1}{\sigma-1}$, such that $\left(
\delta+1\right)  \left(  \sigma-1\right)  =k$, finally yielding the
approximation:%
\begin{multline*}
(\sigma-1)\ln P_{i}\widehat{P}_{j}\\
=\frac{1}{\delta+1}\left\{  -\sum_{l\in I_{j}}s_{l}^{I_{j}}\sum_{h\in J_{i}%
}s_{h}^{J_{i}}\left[  \gamma d_{lh}-\delta\widetilde{z}_{lh}\right]
+\sum_{h\in J_{i}}s_{h}^{J_{i}}\left[  \gamma d_{ih}-\delta\widetilde{z}%
_{ih}\right]  +\sum_{l\in I_{j}}s_{l}^{I_{j}}\left[  \gamma d_{lj}%
-\delta\widetilde{z}_{lj}\right]  \right\}  .
\end{multline*}


\subsubsection{Proof of Implication 1: Bilateral
changes\label{App_prop_bi_proof}}

\begin{proof}
The trade elasticity for a bilateral change is given by%
\begin{align}
\xi_{ij}^{B}  &  =\left(  \sigma-1\right)  +\varphi_{ij}\left(  \sigma
-1\right)  -\left(  1+\varphi_{ij}\right)  \frac{\partial\ln\left(
P_{i}\widehat{P}_{j}\right)  ^{\sigma-1}}{\partial d_{ij,ji}},\nonumber\\
\frac{\partial\ln\left(  P_{i}\widehat{P}_{j}\right)  ^{\sigma-1}}{\partial
d_{ij,ji}}  &  =\left(  \sigma-1\right)  \left(  -s_{i}^{I_{j}}s_{j}^{J_{i}%
}-s_{j}^{I_{j}}s_{i}^{J_{i}}+s_{j}^{J_{i}}+s_{i}^{I_{j}}\right)
,\label{MRbi}\\
&  \Rightarrow\xi_{ij}^{B}=\left(  \sigma-1\right)  \left(  1+\varphi
_{ij}\right)  \left(  1+s_{i}^{I_{j}}s_{j}^{J_{i}}+s_{j}^{I_{j}}s_{i}^{J_{i}%
}-s_{j}^{J_{i}}-s_{i}^{I_{j}}\right)  .\nonumber
\end{align}
or $\xi_{ij}^{B}=\gamma\left(  1+\varphi_{ij}\right)  \left(  1+s_{i}^{I_{j}%
}s_{j}^{J_{i}}+s_{j}^{I_{j}}s_{i}^{J_{i}}-s_{j}^{J_{i}}-s_{i}^{I_{j}}\right)
$ if $\gamma=\sigma-1$. For a change in the size of importer $i$ such that
$\partial s_{i}^{J_{i}}=\partial s_{i}^{I_{j}}=\partial s_{i}$,%
\[
\frac{\partial\xi_{ij}^{B}}{\partial s_{i}}=\frac{\partial\xi_{ij}^{gross}%
}{\partial s_{i}}\left(  1+s_{i}^{I_{j}}s_{j}^{J_{i}}+s_{j}^{I_{j}}%
s_{i}^{J_{i}}-s_{j}^{J_{i}}-s_{i}^{I_{j}}\right)  +\xi_{ij}^{gross}\left(
s_{j}^{J_{i}}+s_{j}^{I_{j}}-1\right)  .
\]
Use Lemma 1 to find%
\[
\frac{\partial\xi_{ij}^{B}}{\partial s_{i}}<0\text{ if }s_{j}^{J_{i}}%
+s_{j}^{I_{j}}<1.
\]
For a change in the size of exporter $j$\ such that $\partial s_{j}^{J_{i}%
}=\partial s_{j}^{I_{j}}=\partial s_{j}$,%
\[
\frac{\partial\xi_{ij}^{B}}{\partial s_{j}}=\frac{\partial\xi_{ij}^{gross}%
}{\partial s_{j}}\left(  1+s_{i}^{I_{j}}s_{j}^{J_{i}}+s_{j}^{I_{j}}%
s_{i}^{J_{i}}-s_{j}^{J_{i}}-s_{i}^{I_{j}}\right)  +\xi_{ij}^{gross}\left(
s_{i}^{I_{j}}+s_{i}^{J_{i}}-1\right)  ,
\]
Use Lemma 1 to find%
\[
\frac{\partial\xi_{ij}^{B}}{\partial s_{j}}<0\text{ if }s_{i}^{I_{j}}%
+s_{i}^{J_{i}}<1.
\]

\end{proof}

\subsubsection{Proof of Implication 2: Multilateral
Changes\label{App_prop_mult_proof}}

\begin{proof}
The trade elasticity for a multilateral change is given by%
\begin{align}
\xi_{ij}^{M}  &  =\left(  \sigma-1\right)  +\varphi_{ij}\left(  \sigma
-1\right)  -\left(  1+\varphi_{ij}\right)  \frac{\partial\ln\left(
P_{i}\widehat{P}_{j}\right)  ^{\sigma-1}}{\partial d},\nonumber\\
\frac{\partial\ln\left(  P_{i}\widehat{P}_{j}\right)  ^{\sigma-1}}{\partial
d}  &  =\left(  \sigma-1\right)  \left[  -\sum_{l\in I_{j}}s_{l}^{I_{j}%
}\left(  1-s_{l}^{J_{i}}\right)  +\left(  1-s_{i}^{J_{i}}\right)  +\left(
1-s_{j}^{I_{j}}\right)  \right] \label{a}\\
&  =\left(  \sigma-1\right)  \left(  1+\sum_{l\in I_{j}}s_{l}^{I_{j}}%
s_{l}^{J_{i}}-s_{i}^{J_{i}}-s_{j}^{I_{j}}\right)  ,\label{b}\\
\Rightarrow\xi_{ij}^{M}  &  =\left(  \sigma-1\right)  +\varphi_{ij}\left(
\sigma-1\right)  -\left(  1+\varphi_{ij}\right)  \left(  \sigma-1\right)
\left(  1+\sum_{l\in I_{j}}s_{l}^{I_{j}}s_{l}^{J_{i}}-s_{i}^{J_{i}}%
-s_{j}^{I_{j}}\right)  . \label{c}%
\end{align}
The second line follows from the fact that, for multilateral changes,
$\partial d_{ij}=\partial d$, $i\neq j$, $\partial d_{ii}=0\,$, such that the
component of MR\ due to world trade resistance $-\sum_{l\in I_{j}}s_{l}%
^{I_{j}}\sum_{h\in J_{i}}s_{h}^{J_{i}}\left[  \gamma d_{lh}-\delta
\widetilde{z}_{lh}\right]  $ changes according to%
\begin{multline*}
-\gamma\left(  1+\delta\right)  \sum_{l\in I_{j}}s_{l}^{I_{j}}\left[
s_{1}^{J_{i}}\partial d_{l1}+s_{2}^{J_{i}}\partial d_{l2}+...+\left(
s_{l}^{J_{i}}\times0\right)  +...+s_{n}^{J_{i}}\partial d_{ln}\right] \\
=-\gamma\left(  1+\delta\right)  \partial d\sum_{l\in I_{j}}s_{l}^{I_{j}%
}\left(  1-s_{l}^{J_{i}}\right)  ,
\end{multline*}
generating the term $\sum_{l\in I_{j}}s_{l}^{I_{j}}\left(  1-s_{l}^{J_{i}%
}\right)  =1-\sum_{l\in I_{j}}s_{l}^{I_{j}}s_{l}^{J_{i}}$ in (\ref{a}) and
(\ref{b}) above. Then (\ref{c}) can be simplified to yield:%
\[
\xi_{ij}^{M}=\left(  \sigma-1\right)  \left(  1+\varphi_{ij}\right)  \left(
-\sum_{l\in I_{j}}s_{l}^{I_{j}}s_{l}^{J_{i}}+s_{i}^{J_{i}}+s_{j}^{I_{j}%
}\right)  ,
\]
or $\xi_{ij}^{M}=\gamma\left(  1+\varphi_{ij}\right)  \left(  -\sum_{l\in
I_{j}}s_{l}^{I_{j}}s_{l}^{J_{i}}+s_{i}^{J_{i}}+s_{j}^{I_{j}}\right)  $ if
$\gamma=\sigma-1.$ Then:

(a) Ignoring MR gives the gross elasticity as $\gamma\left(  1+\varphi
_{ij}\right)  >0$ but including the MR term $\left(  -\sum_{l\in I_{j}}%
s_{l}^{I_{j}}s_{l}^{J_{i}}+s_{i}^{J_{i}}+s_{j}^{I_{j}}\right)  $ implies
$\xi_{ij}^{M}\gtrless0$ since $\left(  -\sum_{l\in I_{j}}s_{l}^{I_{j}}%
s_{l}^{J_{i}}+s_{i}^{J_{i}}+s_{j}^{I_{j}}\right)  \gtrless0$.

(b) (i) If the extensive margin does not operate, the elasticity is $\left.
\xi_{ij}^{M}\right\vert _{\varphi_{ij}=0}=\gamma\left(  -\sum_{l\in I_{j}%
}s_{l}^{I_{j}}s_{l}^{J_{i}}+s_{i}^{J_{i}}+s_{j}^{I_{j}}\right)  $. Then as
importer size changes $\partial s_{i}=\partial s_{i}^{J_{i}}=\partial
s_{i}^{I_{j}}$, $\frac{\partial\left.  \xi_{ij}^{M}\right\vert _{\varphi
_{ij}=0}}{\partial s_{i}}=\gamma\left(  1-s_{i}^{I_{j}}-s_{i}^{J_{i}}\right)
,$ which is positive if $s_{i}^{I_{j}}+s_{i}^{J_{i}}<1$, and as exporter size
changes $\partial s_{j}=\partial s_{j}^{I_{j}}=\partial s_{j}^{J_{i}}$,
$\frac{\partial\left.  \xi_{ij}^{M}\right\vert _{\varphi_{ij}=0}}{\partial
s_{j}}=\gamma\left(  1-s_{j}^{I_{j}}-s_{j}^{J_{i}}\right)  $, which is
positive if $s_{j}^{I_{j}}+s_{j}^{J_{i}}<1$.

(b) (ii) If the extensive margin operates, the sign of $\frac{\partial\xi
_{ij}^{M}}{\partial s_{i}}$ is subject to two competing effects. For changes
in importer size:%
\[
\frac{\partial\xi_{ij}^{M}}{\partial s_{i}}=\underset{-\text{ve}%
}{\underbrace{\frac{\partial\xi_{ij}^{gross}}{\partial s_{i}}}}%
\underset{+\text{ve or }-\text{ve}}{\underbrace{\left(  -\sum_{l\in I_{j}%
}s_{l}^{I_{j}}s_{l}^{J_{i}}+s_{i}^{J_{i}}+s_{j}^{I_{j}}\right)  }%
}+\underset{+\text{ve}}{\underbrace{\xi_{ij}^{gross}\left(  1-s_{i}^{I_{j}%
}-s_{i}^{J_{i}}\right)  }}.
\]
The MR effect from (b)\ (i) increases the elasticity for larger countries, but
larger countries experience smaller direct effects of trade costs on the
extensive margin as shown in Lemma 1. The response of the overall elasticity
to changes in importer size is therefore ambiguous so that $\frac{\partial
\xi_{ij}^{M}}{\partial s_{i}}\lessgtr0.$ An analogous result can be shown for
exporter size, using b(i) above.
\end{proof}

\subsection{Empirical implementation}

\subsubsection{Description of variables\label{App_describe_vars}}

The dependent variable is the log of exports in 1986 measured in constant
(2000) US dollars. The main variables used are distance: the log of distance
in km between the importer and exporter; border: a binary variable indicating
whether the country pair shares a common physical boundary; island: a binary
variable indicating whether at least one country is an island (in HMR, this is
described as an indicator of whether both countries are an island);
landlocked: a binary variable indicating whether at least one country is
landlocked (in HMR, this is described as an indicator of whether both
countries are landlocked); legal: a binary variable indicating whether or not
the country pair share the same legal origin; language: a binary variable
indicating whether the country pair share a common language; colonial ties: a
binary variable indicating whether one country every colonised the other; FTA:
a binary variable indicating whether or not the country pair formed a regional
trade agreement; religion: a variable constructed by HMR indicating how
similar the religious composition is in the country pair (\% Protestants in
$j$\ multiplied by \% Protestants in $i$\ + \% Catholics in $j$\ multiplied by
\% Catholics in $i$\ + \% Muslims in $j$\ multiplied by \% Muslims in
$i$);\footnote{We use the version employed by HMR in their paper. An updated
version has subsequently become available on Elhanan Helpman's website but
does not affect any of our results.}exporter and importer dummies to capture
fixed effects.

To construct GDP Shares, GDP data are sourced from the World Bank's World
Development Indicators. Where necessary, WDI data were combined to meet the
country definitions in the trade data (for example combining Belgium and
Luxembourg). Where possible, missing observations were supplemented with data
from the United Nations Common Database (UNCDB). Otherwise, a GDP of 0.1 was
inputted manually. The denominator is constructed based on the sum of the
individual countries' GDPs. Shares are based on the subset of countries the
importer imports from or the exporter exports to. Except in the example of a
small exporter with few trading partners exporting to a large exporter, we
have achieved similar results at the bilateral level simply by using the share
of world GDP, where world GDP is the value provided by the WDI.

\subsubsection{Description of empirical methodology\label{App_describe_method}%
}

Here, we describe the methodology used for estimation and the comparative
statics exercises.

\paragraph{Estimation}

\begin{enumerate}
\item Estimate a probit model for the probability of positive exports from $j
$\ to $i$. The first stage includes the bilateral variables listed in the data
appendix together with importer and exporter dummies. The set of bilateral
variables must include a variable which will be omitted from the second stage.
Theoretically, such a variable should affect the fixed costs of exporting, but
not variable costs. For the full sample, HMR use religion for this
purpose.\footnote{Some countries export to everyone else or import from
everyone else. The fixed effects are thus perfect predictors and cannot be
included in the probit. HMR omit such variables from the study. However, an
alternative option might be to generate predicted probabilities equal to
$0.9999999$, as done in step 2 for a number of countries, and continue from
there.}

\item Generate predicted probabilities ($\hat{\rho}$) of positive exports.
Many of these are practically indistinguishable from unity such that many
values of the Inverse Mills Ratios are not distinguished from each other. The
approach in HMR converts all $\hat{\rho}>0.9999999$\ to values of $\hat{\rho
}=0.9999999$.

\item Predict the Inverse Mills Ratio $\widehat{\bar{\eta}}_{ij}^{\ast}%
=\frac{\phi(\widehat{z}_{ij}^{\ast})}{\Phi(\widehat{z}_{ij}^{\ast})}$, where
$\phi(.)$\ is the the standard normal density function and $\Phi(.)$\ is the
standard normal distribution function. Generate $\widehat{z}_{ij}^{\ast}$ as a
linear function of the probit variables and $\widehat{\bar{z}}_{ij}^{\ast}=$
$\widehat{z}_{ij}^{\ast}+$ \ $\widehat{\bar{\eta}}_{ij}^{\ast}$.

\item Estimate the second stage including the predicted controls for firm
selection $\log(e^{\delta\widehat{\bar{z}}_{ij}^{\ast}}-1)$\ and
country-selection $\widehat{\bar{\eta}}_{ij}^{\ast}$. HMR07 use maximum
likelihood while HMR08 use non-linear least squares (NLS). We use NLS. Because
of the large number of dummies (over 300), we follow Carayol (2006) and use
the function evaluator program version of the nl command in \textit{Stata}.
Our results, while extremely close, do not precisely match HMR and are in fact
closer to those in HMR07. Estimates are presented in the accompanying table
for convenience.

\item Generate predicted values for trade $\hat{m}$\ including a predicted
value $w=\log(e^{\hat{\delta}\widehat{\bar{z}}^{\ast}}-1)$.
\end{enumerate}

\paragraph{Simulations}

Generate an alternative measure of the variable of interest.\ In the case of
distance, the new measure is 10\% lower than the original.

\begin{enumerate}
\item Generate a new latent variable using estimated coefficients together
with the new distance measure.

\item Generate the new predicted probabilities $\hat{\rho}^{\prime}$, setting
values above $0.9999999$\ equal to $0.9999999$\ as before.

\item Keep the originally estimated Inverse Mills Ratio $\widehat{\bar{\eta}%
}_{ij}^{\ast}$. The original estimate of unobserved trade frictions,
conditional on the same countries trading, is still the prediction based on
the original values.

\item Generate new predictions $\widehat{\bar{z}}^{\ast\prime}$, using the
same value for $\widehat{\bar{\eta}}_{ij}^{\ast}$\ but the new predicted
probabilities in $\Phi^{-1}(\hat{\rho})$. Use the estimate $\hat{\delta}$\ and
$\widehat{\bar{z}}^{\ast\prime}$\ to generate an alternative prediction of the
non-linear term $w^{\prime}=\log(e^{\hat{\delta}\widehat{\bar{z}}^{\ast\prime
}}-1)$.

\item Generate the alternative predicted values for trade $\hat{m}^{\prime}%
$\ based on the new distance values and $w^{\prime}$.

\item Calculate the gross elasticity $\xi=\frac{\hat{m}^{\prime}-\hat{m}%
}{\left\vert \log0.9\right\vert }.$\ For country-pairs with $\hat{\rho
}^{\prime}>0.9999999$, assign the elasticity generated by a value of
$\hat{\rho}^{\prime}=0.9999999$. By doing this, HMR are taking a probability
of $0.9999999$\ to imply a certain proportion of firms exporting.

\item Calculate the bilateral net elasticities applying the appropriate
country shares adjusting for trading partners using either $\xi_{ij}%
^{B}=\gamma+\varphi_{ij}\gamma_{p}-\left(  1+\varphi_{ij}\right)  \frac
{\gamma+\delta\gamma_{p}}{1+\delta}\left(  -s_{i}^{I_{j}}s_{j}^{J_{i}}%
-s_{j}^{I_{j}}s_{i}^{J_{i}}+s_{j}^{J_{i}}+s_{i}^{I_{j}}\right)  $ or $\xi
_{ij}^{M}=\gamma+\varphi_{ij}\gamma_{p}-\left(  1+\varphi_{ij}\right)
\frac{\gamma+\delta\gamma_{p}}{1+\delta}\left(  1+\sum_{l\in I_{j}}%
s_{l}^{I_{j}}s_{l}^{J_{i}}-s_{i}^{J_{i}}-s_{j}^{I_{j}}\right)  $. By ignoring
the potential difference in estimates of $\gamma$ and $\gamma_{p}$, one could
simply attain the desired elasticity by multiplying the gross elasticity by
the share terms.
\end{enumerate}

\newpage

\subsection{Regression results}

\setcounter{table}{0} \numberwithin{table}{section}

Table \ref{results_table} contains the regression results. The first column
gives the results from standard OLS estimates ignoring firm heterogeneity.
Column 2 gives first-stage probit estimates and column 3 gives the
second-stage regression, which includes controls for country selection (the
Inverse Mills Ratio $\widehat{\overline{\eta}}_{ij}^{\ast}$) and the
proportion of firms exporting (represented by the non-linear parameter
$\delta$). In all cases, multilateral resistance is controlled for in
estimation by the use of exporter and importer fixed effects. For
completeness, column 4 includes results based on the semi-paramteric
polynomial approximation for the proportion of firms exporting.

\begin{table}[ptb]
\caption{Estimated gravity equations. Dependent variable is bilateral trade
flows. }%
\label{results_table}
\onehalfspacing
\begin{threeparttable}
\centering
\setlength{\tabcolsep}{12pt}
\begin{tabular}{ccccc}
& (1) OLS &  (2) Probit
&  (3) NL &  (4) Polynomial \\ \hline\hline
\multicolumn{1}{l}{\textit{Distance (}$-\gamma $\textit{)}} & -1.176*** & -0.660*** & -0.799*** & -0.862*** \\
\multicolumn{1}{l}{} &  (0.0269) &  (0.0206) &  (0.0352) &  (0.038) \\
\multicolumn{1}{l}{\textit{Border}} & 0.458*** & -0.382*** & 0.832*** & 0.786*** \\
\multicolumn{1}{l}{} &  (0.118) &  (0.0941) &  (0.124) &  0.149 \\
\multicolumn{1}{l}{\textit{Island}} & -0.391*** & -0.345*** & -0.17 & -0.2 \\
\multicolumn{1}{l}{} &  (0.107) &  (0.0751) &  (0.097) &  0.12 \\
\multicolumn{1}{l}{\textit{Landlocked}} & -0.561*** & -0.181 & -0.447*** & -0.482 \\
\multicolumn{1}{l}{} &  (0.164) &  (0.105) &  (0.128) &  0.135 \\
\multicolumn{1}{l}{\textit{Legal}} & 0.486*** & 0.0964** & 0.388*** & 0.385*** \\
\multicolumn{1}{l}{} &  (0.043) &  (0.0297) &  (0.039) &  0.044 \\
\multicolumn{1}{l}{\textit{Language}} & 0.176** & 0.284*** & 0.023 & 0.045 \\
\multicolumn{1}{l}{} &  (0.0556) &  (0.0381) &  (0.059) &  0.066 \\
\multicolumn{1}{l}{\textit{Colonial ties}} & 1.299*** & 0.325 & 1.003*** & 1.038*** \\
\multicolumn{1}{l}{} &  (0.132) &  (0.292) &  (0.091) &  0.092 \\
\multicolumn{1}{l}{\textit{Currency Union}} & 1.364*** & 0.492*** & 1.024*** & 1.106*** \\
\multicolumn{1}{l}{} &  (0.216) &  (0.132) &  (0.269) &  0.249 \\
\multicolumn{1}{l}{\textit{FTA}} & 0.759*** & 1.985*** & 0.378** & 0.457***\\
\multicolumn{1}{l}{} &  (0.162) &  (0.314) &  (0.144) &  0.108 \\
\multicolumn{1}{l}{\textit{Religion}} & 0.102 & 0.261*** &  &  \\
\multicolumn{1}{l}{} &  (0.09) &  (0.061) &  &  \\
\multicolumn{1}{l}{$\widehat{\overline{\eta }}^{\ast }$} &  &  & 0.392*** & 1.131*** \\
\multicolumn{1}{l}{} &  &  &  (0.049) &  0.137 \\
\multicolumn{1}{l}{$\widehat{\delta }$} &  &  & 0.716*** &  \\
\multicolumn{1}{l}{} &  &  &  (0.049) &  \\
\multicolumn{1}{l}{$\widehat{z}^{\ast }$} &  &  &  & 3.602*** \\
\multicolumn{1}{l}{} &  &  &  &  0.348 \\
\multicolumn{1}{l}{$\widehat{z}^{\ast 2}$} &  &  &  & -0.782 \\
\multicolumn{1}{l}{} &  &  &  &  0.112 \\
\multicolumn{1}{l}{$\widehat{z}^{\ast 3}$} &  &  &  & 0.064*** \\
\multicolumn{1}{l}{} &  &  &  &  0.011 \\
\multicolumn{1}{l}{\textit{Constant}} & 3.168*** & 3.813*** & 517.2 & 4.197*** \\
\multicolumn{1}{l}{} &  (0.788) &  (0.371) &  (764.8) &  0.517 \\ \hline
\multicolumn{1}{l}{ Observations  [$R^{2}$ ]} &  11,146  [0.709] &  24,649 &  11,146  [0.718] &  11,146 [0.721] \\ \hline \hline
\end{tabular}
\begin{tablenotes}
\item \textit{Notes:} Significance at 0.001 \& 0.01 levels denoted with `***' and `**'
respectively. Standard errors in parentheses (based on 50 bootstrap
replications in columns 3 and 4).
\end{tablenotes}
\end{threeparttable}
\end{table}

\begin{thebibliography}{9}                                                                                                %
\bibitem {}Carayol, T. (2006), `"Expression too long" error with nl command', Statalist
\end{thebibliography}


\end{document}